| L(s) = 1 | + (1.68 + 1.08i)2-s + (0.426 + 2.96i)3-s + (1.66 + 3.63i)4-s + (16.0 + 4.71i)5-s + (−2.49 + 5.45i)6-s + (17.1 − 19.7i)7-s + (−1.13 + 7.91i)8-s + (−8.63 + 2.53i)9-s + (21.9 + 25.3i)10-s + (2.91 − 1.87i)11-s + (−10.0 + 6.48i)12-s + (−15.3 − 17.6i)13-s + (50.2 − 14.7i)14-s + (−7.15 + 49.7i)15-s + (−10.4 + 12.0i)16-s + (−29.3 + 64.2i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (0.0821 + 0.571i)3-s + (0.207 + 0.454i)4-s + (1.43 + 0.422i)5-s + (−0.169 + 0.371i)6-s + (0.926 − 1.06i)7-s + (−0.0503 + 0.349i)8-s + (−0.319 + 0.0939i)9-s + (0.693 + 0.800i)10-s + (0.0798 − 0.0513i)11-s + (−0.242 + 0.156i)12-s + (−0.327 − 0.377i)13-s + (0.959 − 0.281i)14-s + (−0.123 + 0.856i)15-s + (−0.163 + 0.188i)16-s + (−0.418 + 0.916i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.56719 + 1.54757i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.56719 + 1.54757i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.68 - 1.08i)T \) |
| 3 | \( 1 + (-0.426 - 2.96i)T \) |
| 23 | \( 1 + (80.0 - 75.9i)T \) |
| good | 5 | \( 1 + (-16.0 - 4.71i)T + (105. + 67.5i)T^{2} \) |
| 7 | \( 1 + (-17.1 + 19.7i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (-2.91 + 1.87i)T + (552. - 1.21e3i)T^{2} \) |
| 13 | \( 1 + (15.3 + 17.6i)T + (-312. + 2.17e3i)T^{2} \) |
| 17 | \( 1 + (29.3 - 64.2i)T + (-3.21e3 - 3.71e3i)T^{2} \) |
| 19 | \( 1 + (2.36 + 5.18i)T + (-4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (-48.6 + 106. i)T + (-1.59e4 - 1.84e4i)T^{2} \) |
| 31 | \( 1 + (-14.9 + 103. i)T + (-2.85e4 - 8.39e3i)T^{2} \) |
| 37 | \( 1 + (369. - 108. i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (324. + 95.2i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (31.1 + 216. i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 - 452.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-389. + 449. i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (-81.6 - 94.1i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (83.4 - 580. i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (587. + 377. i)T + (1.24e5 + 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-118. - 75.8i)T + (1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (265. + 580. i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (-572. - 660. i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (-1.02e3 + 301. i)T + (4.81e5 - 3.09e5i)T^{2} \) |
| 89 | \( 1 + (225. + 1.57e3i)T + (-6.76e5 + 1.98e5i)T^{2} \) |
| 97 | \( 1 + (558. + 164. i)T + (7.67e5 + 4.93e5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.49853618634949596604554937490, −11.87829654831272361750246973633, −10.57802627896553434625121344674, −10.12500266249817623013929155017, −8.643525939184809751343029632282, −7.36862143346094180490467127135, −6.12107386920489536086170632075, −5.08165791723220426800236563128, −3.80713812410867244497200360952, −2.01372285356549819354482400408,
1.63671750343127539229820962756, 2.50839917835684383741928393564, 4.87331714346208035350126560703, 5.65385023003951671819491596484, 6.81751986822928233986385880951, 8.547057765453851240357809979302, 9.361869101293874920176700606259, 10.61053198296985118556057788627, 11.92457629093581425336654845713, 12.43256302927186563178429533665
